Control theorems for fine Selmer groups, and duality of fine Selmer groups attached to modular forms

TL;DR

This paper establishes control theorems for fine Selmer groups of modules over local rings and explores their duality properties in the context of modular forms over cyclotomic extensions.

Contribution

It introduces new control theorems for fine Selmer groups and compares Selmer groups attached to modular forms and their conjugates in cyclotomic extensions.

Findings

Proves control theorems for fine Selmer groups of cofinitely generated modules.

Establishes duality relations for fine Selmer groups attached to modular forms.

Provides a framework for comparing Selmer groups of modular forms and their conjugates.

Abstract

Let $\mathcal{O}$ be the ring of integers of a finite extension of $\mathbb{Q}_p$. We prove two control theorems for fine Selmer groups of general cofinitely generated modules over $\mathcal{O}$. We apply these control theorems to compare the fine Selmer group attached to a modular form $f$ over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ to its counterpart attached to the conjugate modular form $\overline{f}$.